Let $A, B \in L(V)$ (or equivalently $n \times n$ matrix), where $V$ is a $n$-dimensional vector space. There are a multiple of proofs why $\det(A)\det(B) = \det(AB)$, but I couldn't find a satisfactory that is intuition-appealing.
First, the most imminent motivation for determinant is volume: $\det(A)$ is oriented volume of parallelepiped consisting $n$-column vectors of $A$. Given three matrices $A, B, AB$, we have three distinct parallelepiped each of which has oriented volume $\det(A)$, $\det(B)$, and $\det(AB)$, respectively.
Second, the given $A, B \in L(V)$ as in linear transformation, $AB$ is simply a composition of linear transformation.
I am trying to relate these two concepts, but it feels that the gap between two concepts are too broad. Is there a nice interpretation that fills the gap between these two concepts ?